TIFR CSE 2021 | Part A | Question: 8

Question

Consider the sequence

$$y_{n}=\frac{1}{\int_{1}^{n}\frac{1}{\left ( 1+x/n \right )^{3}}dx}$$

for $\text{n} = 2,3,4, \dots$ Which of the following is $\text{TRUE}$?

• A. The sequence $\{y_{n}\}$ does not have a limit as $n\rightarrow \infty$.
• B. $y_{n}\leq 1$ for all $\text{n} = 2,3,4, \dots$
• C. $\lim_{n\rightarrow \infty }y_{n}$ exists and is equal to $6/\pi ^{2}$.
• D. $\lim_{n\rightarrow \infty } y_{n}$ exists and is equal to $0$.
• E. The sequence $\{y_{n}\}$ first increases and then decreases as $\text{n}$ takes values $2, 3, 4, \dots$